List of paradoxes

This is a list of paradoxes, grouped thematically. The grouping is approximate, as paradoxes may fit into more than one category. Because of varying definitions of the term paradox, some of the following are not considered to be paradoxes by everyone. This list collects only scenarios that have been called a paradox by at least one source and have their own article.

Although considered paradoxes, some of these are based on fallacious reasoning, or incomplete/faulty analysis. Informally, the term is often used to describe a counter-intuitive result.

Logic

Barbershop paradox The supposition that if one of two simultaneous assumptions leads to a contradiction, the other assumption is also disproved leads to paradoxical consequences. Not to be confused with the Barber paradox.

What the Tortoise Said to Achilles "Whatever Logic is good enough to tell me is worth writing down...", also known as Carroll's paradox, not to be confused with the physical paradox of the same name.

Catch-22 A situation in which someone is in need of something that can only be had by not being in need of it. A soldier who wants to be declared insane in order to avoid combat is deemed not insane for that very reason, and will therefore not be declared insane.

Drinker paradox In any pub there is a customer of whom it is true to say: if that customer drinks, everybody in the pub drinks.

Lottery paradox If there is one winning ticket in a large lottery, it is reasonable to believe of any particular lottery ticket that it is not the winning ticket, but it is not reasonable to believe that no lottery ticket will win.

Self-reference

These paradoxes have in common a contradiction arising from either from self-reference or from circular reference, in which several statements refer to each other in a way that following some of the references leads back to the starting point.

Bhartrhari's paradox The thesis that there are some things which are unnameable conflicts with the notion that something is named by calling it unnameable.

Berry paradox The phrase "the first number not nameable in under ten words" appears to name it in nine words.

Crocodile dilemma If a crocodile steals a child and promises its return if the father can correctly guess exactly what the crocodile will do, how should the crocodile respond in the case that the father guesses that the child will not be returned?

Paradox of the Court A law student agrees to pay his teacher after (and only after) winning his first case. The teacher then sues the student (who has not yet won a case) for payment.

Quine's paradox "'Yields a falsehood when appended to its own quotation' yields a falsehood when appended to its own quotation." Shows that a sentence can be paradoxical even if it is not self-referring and does not use demonstratives or indexicals.

Yablo's paradox An ordered infinite sequence of sentences, each of which says that all following sentences are false. While constructed to avoid self-reference, there is no consensus whether it relies on self-reference or not.

Opposite Day "It is opposite day today." Therefore it is not opposite day, but if you say it is a normal day it would be considered a normal day.

Vagueness

Ship of Theseus (a.k.a. George Washington's axe or Grandfather's old axe): It seems like you can replace any component of a ship, and it is still the same ship. So you can replace them all, one at a time, and it is still the same ship. However, you can then take all the original pieces, and assemble them into a ship. That, too, is the same ship you began with.

Sorites paradox (also known as the paradox of the heap): If you remove a single grain of sand from a heap, you still have a heap. Keep removing single grains, and the heap will disappear. Can a single grain of sand make the difference between heap and non-heap?

Lindley's paradox Tiny errors in the null hypothesis are magnified when large data sets are analyzed, leading to false but highly statistically significant results.

Low birth weight paradox Low birth weight and mothers who smoke contribute to a higher mortality rate. Babies of smokers have lower average birth weight, but low birth weight babies born to smokers have a lower mortality rate than other low birth weight babies. This is a special case of Simpson's paradox.

Simpson's paradox, or the Yule–Simpson effect: A trend that appears in different groups of data disappears when these groups are combined, and the reverse trend appears for the aggregate data.

Will Rogers phenomenon The mathematical concept of an average, whether defined as the mean or median, leads to apparently paradoxical results—for example, it is possible that moving an entry from an encyclopedia to a dictionary would increase the average entry length on both books.

Two-envelope paradox You are given two indistinguishable envelopes, each of which contains a positive sum of money. One envelope contains twice as much as the other. You may pick one envelope and keep whatever amount it contains. You pick one envelope at random but before you open it you are given the chance to take the other envelope instead.

Infinity and infinitesimals

Cantor's paradox The set of all sets would have its own power set as a subset, therefore its cardinality would be at least as great as that of its power set. But Cantor's theorem proves that power sets are strictly greater than the sets they are constructed from. Consequently, the set of all sets would contain a subset greater than itself.

Coin rotation paradox a coin rotating along the edge of an identical coin will make a full revolution after traversing only half of the stationary coin's circumference.

Gabriel's Horn or Torricelli's trumpet: A simple object with finite volume but infinite surface area. Also, the Mandelbrot set and various other fractals are covered by a finite area, but have an infinite perimeter (in fact, there are no two distinct points on the boundary of the Mandelbrot set that can be reached from one another by moving a finite distance along that boundary, which also implies that in a sense you go no further if you walk "the wrong way" around the set to reach a nearby point). This can be represented by a Klein bottle.

Hausdorff paradox There exists a countable subset C of the sphere S such that S\C is equidecomposable with two copies of itself.

Nikodym set A set contained in and with the same Lebesgue measure as the unit square, yet for every one of its points there is a straight line intersecting the Nikodym set only in that point.

Decision theory

Abilene paradox People can make decisions based not on what they actually want to do, but on what they think that other people want to do, with the result that everybody decides to do something that nobody really wants to do, but only what they thought that everybody else wanted to do.

Apportionment paradox Some systems of apportioning representation can have unintuitive results due to rounding

Alabama paradox Increasing the total number of seats might shrink one block's seats.

New states paradox Adding a new state or voting block might increase the number of votes of another.

Cool tropics paradox A contradiction between modelled estimates of tropical temperatures during warm, ice-free periods of the Cretaceous and Eocene, and the lower temperatures that proxies suggest were present.

Astrophysics

Algol paradox In some binaries the partners seem to have different ages, even though they are thought to have formed at the same time.

Faint young Sun paradox The contradiction between existence of liquid water early in the Earth's history and the expectation that the output of the young Sun would have been insufficient to melt ice on Earth.

Tea leaf paradox When a cup of tea is stirred, the leaves assemble in the center, even though centrifugal force pushes them outward.

Upstream contamination When a fluid is poured from a higher container onto a lower one, particles can climb up the falling water.

Cosmology

Bentley's paradox In a Newtonian universe, gravitation should pull all matter into a single point.

Boltzmann brain If the universe we observe resulted from a random thermodynamic fluctuation, it would be vastly more likely to be a simple one than the complex one we observe. The simplest case would be just a brain floating in vacuum, having the thoughts and sensations you have.

Fermi paradox If there are, as various arguments suggest, many other sentient species in the Universe, then where are they? Shouldn't their presence be obvious?

Heat death paradox If the universe was infinitely old, it would be in thermodynamical equilibrium, which contradicts what we observe.

Olbers' paradox Why is the night sky dark if there is an infinity of stars, covering every part of the celestial sphere?

Electromagnetism

Faraday paradox An apparent violation of Faraday's law of electromagnetic induction.

Quantum mechanics

Aharonov–Bohm effect a charged particle is affected by an electromagnetic field even though it has no local contact with that field

Extinction paradox In the small wavelength limit, the total scattering cross section of an impenetrable sphere is twice its geometrical cross-sectional area (which is the value obtained in classical mechanics).[2]

Hardy's paradox How can we make inferences about past events that we haven't observed while at the same time acknowledge that the act of observing it affects the reality we are inferring to?

Klein paradox When the potential of a potential barrier becomes similar to the mass of the impinging particle, it becomes transparent.

Health and nutrition

French paradox The observation that the French suffer a relatively low incidence of coronary heart disease, despite having a diet relatively rich in saturated fats, which are assumed to be the leading dietary cause of such disease.

Glucose paradox The large amount of glycogen in the liver cannot be explained by its small glucose absorption.

Hispanic paradox The finding that Hispanics in the U.S. tend to have substantially better health than the average population in spite of what their aggregate socio-economic indicators predict.

Israeli paradox The observation that Israelis suffer a relatively high incidence of coronary heart disease, despite having a diet very low in saturated fats, which are assumed to be the leading dietary cause of such disease.

Meditation paradox The amplitude of heart rate oscillations during meditation was significantly greater than in the pre-meditation control state and also in three non-meditation control groups[6]

Mexican paradox Mexican children tend to have higher birth weights than can be expected from their socio-economic status.

Obesity survival paradox Although the negative health consequences of obesity in the general population are well supported by the available evidence, health outcomes in certain subgroups seem to be improved at an increased BMI.

Peto's paradox Humans get cancer with high frequency, while larger mammals, like whales, do not. If cancer is essentially a negative outcome lottery at the cell level, and larger organisms have more cells, and thus more potentially cancerous cell divisions, one would expect larger organisms to be more predisposed to cancer.

Pulsus paradoxus A pulsus paradoxus is a paradoxical decrease in systolic blood pressure during inspiration. It can indicate certain medical conditions in which there is reduced venous return of blood to the heart, such as cardiac tamponade or constrictive pericarditis. Also known as the Pulse Paradox.[7]

Chemistry

Levinthal paradox The length of time that it takes for a protein chain to find its folded state is many orders of magnitude shorter than it would be if it freely searched all possible configurations.

SAR paradox Exceptions to the principle that a small change in a molecule causes a small change in its chemical behaviour are frequently profound.

Time travel

Bootstrap paradox Can a time traveler send himself information with no outside source?

Polchinski's paradox A billiard ball can be thrown into a wormhole in such a way that it would emerge in the past and knock its incoming past self away from the wormhole entrance, creating a variant of the grandfather paradox.

Predestination paradox A man travels back in time to discover the cause of a famous fire. While in the building where the fire started, he accidentally knocks over a kerosene lantern and causes a fire, the same fire that would inspire him, years later, to travel back in time. The bootstrap paradox is closely tied to this, in which, as a result of time travel, information or objects appear to have no beginning.

Grandfather paradox You travel back in time and kill your grandfather before he conceives one of your parents, which precludes your own conception and, therefore, you couldn't go back in time and kill your grandfather.

Hitler's murder paradox You travel back in time and kill a famous person in history before they become famous; but if the person had never been famous then he could not have been targeted as a famous person.

Linguistics and artificial intelligence

Bracketing paradox Is a "historical linguist" a linguist who is historical, or someone who studies "historical linguistics"?

Economics

One class of paradoxes in economics are the paradoxes of competition, in which behavior that benefits a lone actor would leave everyone worse off if everyone did the same. These paradoxes are classified into circuit, classical and Marx paradoxes.

Allais paradox A change in a possible outcome that is shared by different alternatives affects people's choices among those alternatives, in contradiction with expected utility theory.

Lucas paradox Capital is not flowing from developed countries to developing countries despite the fact that developing countries have lower levels of capital per worker, and therefore higher returns to capital.

Mayfield's paradox Keeping everyone out of an information system is impossible, but so is getting everybody in.

Metzler paradox The imposition of a tariff on imports may reduce the relative internal price of that good.

Paradox of prosperity Why do generations that significantly improve the economic climate seem to generally rear a successor generation that consumes rather than produces?

Paradox of thrift If everyone saves more money during times of recession, then aggregate demand will fall and will in turn lower total savings in the population.

Paradox of toil If everyone tries to work during times of recession, lower wages will reduce prices, leading to more deflationary expectations, leading to further thrift, reducing demand and thereby reducing employment.

Paradox of value, also known as diamond-water paradox: Water is more useful than diamonds, yet is a lot cheaper.

Paradox of Plenty The Paradox of Plenty (resource curse) refers to the paradox that countries and regions with an abundance of natural resources, specifically point-source non-renewable resources like minerals and fuels, tend to have less economic growth and worse development outcomes than countries with fewer natural resources.

Politics

Stability–instability paradox When two countries each have nuclear weapons, the probability of a direct war between them greatly decreases, but the probability of minor or indirect conflicts between them increases.

Psychology and sociology

Gender paradox Women conform more closely than men to sociolinguistics norms that are overtly prescribed, but conform less than men when they are not.

Status paradox Several paradoxes involve the concept of medical or social status.

Stockdale paradox "You must never confuse faith that you will prevail in the end—which you can never afford to lose—with the discipline to confront the most brutal facts of your current reality, whatever they might be."

The Paradox of Anti-Semitism A book arguing that the lack of external persecutions and antagonisms results in the dissolution of Jewish identity, a theory that resonates in works of Dershowitz and Sartre.

Miscellaneous

Ant on a rubber rope An ant crawling on a rubber rope can reach the end even when the rope stretches much faster than the ant can crawl.

Bonini's paradox Models or simulations that explain the workings of complex systems are seemingly impossible to construct. As a model of a complex system becomes more complete, it becomes less understandable, for it to be more understandable it must be less complete and therefore less accurate. When the model becomes accurate, it is just as difficult to understand as the real-world processes it represents.

Notes

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As of 2010[update], an image of Pinocchio with a speech bubble "My nose will grow now!" has become a minor Internet phenomenon (Google search, Google image search). It seems likely that this paradox has been independently conceived multiple times.